MATH1061 (S1, 2023) Assignment 1
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH1061 (S1, 2023)
Assignment 1
Due: 4pm on 20 March 2023
All assignments in this course must be submitted electronically and SUBMITTED AS A SINGLE PDF FILE. Prepare your assignment solutions using Word, LaTeX, Windows Journal, or other application, ensuring that your name, student number and tutorial group number appear clearly at the top to the first page, and then save your file in pdf format. Alternatively, you may handwrite your solutions and scan or photograph your handwritten work to create a pdf file. Make sure that your pdf file is legible and that the file size is not excessive. Use the assignment submission link in Blackboard to submit the pdf file.
1. (3 marks) Let p, q and r be statements. Construct a truth table for the following statement
form and determine whether it is a tautology, a contradiction, or neither of these. (欢 (p ^ (q 一欢 r)) 一 (r 一 (p^ 欢 q))
2. (7 marks) Let p, q and r be statements.
(a) Use the Laws of Logical Equivalence and the equivalence of 一 to a disjunction to show
that:
(p 一 (q 一 r)) V (欢 r) 三 (p 一欢 q) V (欢 r)
Show your working and name the laws that you use at each step.
(b) A MATH1061 student reads part (a) above and remarks,
“Take a close look at the two statements we need to show are logically equivalent. Both statement include V(欢 r) at the end. So it is probably fine to just show that (p 一 (q 一 r)) is logically equivalent to (p 一欢 q).”
Do you agree with this student’s remark? Explain why or why not.
3. (5 marks) Determine whether the following argument is valid. Explain your reasoning.
If I go to work then I brush my teeth or drive my car.
I tie my shoes and I don’t wear socks.
If I wear socks then I don’t brush my teeth.
I go to work.
Therefore I drive my car or I don’t wear shoes.
4. (4 marks) Rewrite each of the following sentences using quantifiers (A, 3) and mathematical
notation (e, Z, V, 一, ...) and state whether the sentence is true or false with a brief reason.
(a) A necessary condition for a real number to be rational is that its square is rational. (b) A sufficient condition for a real number to be rational is that its square is rational.
5. (6 marks) Write down the negation of each of the following statements. Then determine whether the statement or its negation is true, and explain why.
(a) Aq e Q, 3r e Q such that e Z.
(b) 3z e Z such that Ax,y e Z, x _ y < z .
6. (5 marks) The following question appeared on an assessment item in MATH1061/7861 in a previous year.
Consider the following statement. “There exists an integer a such that for all integers b, ab > 0.”Write down the negation of this statement. Then determine whether the statement or its negation is true, and explain why.
In February 2023, a lecturer posed this question to a generative AI model and the answer it produced is given below. Provide a careful critique of this answer, including correcting any mistakes it may have made.
——————————————————————————–
The given statement is:
3a ∈ Z, Ab ∈ Z,ab > 0
The negation of this statement would be:
Aa ∈ Z, 3b ∈ Z,ab ≤ 0
In words, the negation states that for all integers a, there exists an integer b such that ab is less than or equal to 0 .
Now let’s consider whether the original statement or its negation is true .
The original statement asserts the existence of an integer a such that the product of a and any integer b is greater than 0 . This is true if a is positive, because in that case, the product of a and any positive or negative integer will be greater than 0 . Therefore, the original statement is true for all positive integers a .
On the other hand, the negation of the statement asserts that for every integer a, there exists an integer b such that the product of a and b is less than or equal to 0 . This is true for all values of a, because we can simply choose b = 0, and then ab = 0, which is less than or equal to 0 .
Therefore, the original statement is true and its negation is false .
——————————————————————————–
2023-03-16